Regular circle actions on 2-connected 7-manifolds
The author studies the existence of regular circle actions on closed, oriented \(2\)-connected \(7\)-manifolds. The manifolds are assumed to be either smooth or topological. The results are stated in terms of a certain family \(M^c_{l,k}\), \(c\in \{0,1\}\), \(l,k\in{\mathbb{Z}}\), of \(2\)-connected \(7\)-manifolds. The manifolds \(M^0_{l,k}\) are completely classified in [\textit{D. Crowley} and \textit{C. Escher}, Differ. Geom. Appl. 18, No. 3, 363--380 (2003; Zbl 1027.55014)]. The manifold \(M^1_{l,k}\) is a non-smoothable topological manifold homotopy equivalent to \(M^0_{l,k}\) when \(k \equiv 0\mod 2\). The group \(\Gamma_7\) of exotic \(7\)-spheres is cyclic of order \(28\) with generator \(M^0_{1,1}\). Let \(\Sigma_r= rM^0_{1,1}\in\Gamma_7\), \(r\in {\mathbb{Z}}\). The main result is the following: All homeomorphism classes of the \(2\)-connected \(7\)-manifolds that admit regular circle actions are represented by the connected sums NEWLINE\[NEWLINE M= \#_{2r} S^3\times S^4\# M^c_{6m, (1+c)k}, NEWLINE\]NEWLINE where \(c\in \{ 0,1\}\), \(r\in{\mathbb{N}}\) and \(m,k\in{\mathbb{Z}}\), and \(M\) is smoothable if and only if \(c=0\). All diffeomorphism classes of the smooth \(2\)-connected \(7\)-manifolds that admit smooth regular circle actions are represented by NEWLINE\[NEWLINE \#_{2r}S^3\times S^4\# M^0_{6(a+1)m,(a+1)k}\#\Sigma_{(1-a)m}, NEWLINE\]NEWLINE where \(a\in \{ 0,1\}\), \(r\in{\mathbb{N}}\), \(m,k\in{\mathbb{Z}}\).
scientific article; zbMATH DE number 6355576
scientific article